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Mirrors > Home > MPE Home > Th. List > Mathboxes > 6gbe | Structured version Visualization version GIF version |
Description: 6 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.) |
Ref | Expression |
---|---|
6gbe | ⊢ 6 ∈ GoldbachEven |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6even 44229 | . 2 ⊢ 6 ∈ Even | |
2 | 3prm 16028 | . . 3 ⊢ 3 ∈ ℙ | |
3 | 3odd 44226 | . . . 4 ⊢ 3 ∈ Odd | |
4 | gbpart6 44284 | . . . 4 ⊢ 6 = (3 + 3) | |
5 | 3, 3, 4 | 3pm3.2i 1336 | . . 3 ⊢ (3 ∈ Odd ∧ 3 ∈ Odd ∧ 6 = (3 + 3)) |
6 | eleq1 2877 | . . . . 5 ⊢ (𝑝 = 3 → (𝑝 ∈ Odd ↔ 3 ∈ Odd )) | |
7 | biidd 265 | . . . . 5 ⊢ (𝑝 = 3 → (𝑞 ∈ Odd ↔ 𝑞 ∈ Odd )) | |
8 | oveq1 7142 | . . . . . 6 ⊢ (𝑝 = 3 → (𝑝 + 𝑞) = (3 + 𝑞)) | |
9 | 8 | eqeq2d 2809 | . . . . 5 ⊢ (𝑝 = 3 → (6 = (𝑝 + 𝑞) ↔ 6 = (3 + 𝑞))) |
10 | 6, 7, 9 | 3anbi123d 1433 | . . . 4 ⊢ (𝑝 = 3 → ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (𝑝 + 𝑞)) ↔ (3 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (3 + 𝑞)))) |
11 | biidd 265 | . . . . 5 ⊢ (𝑞 = 3 → (3 ∈ Odd ↔ 3 ∈ Odd )) | |
12 | eleq1 2877 | . . . . 5 ⊢ (𝑞 = 3 → (𝑞 ∈ Odd ↔ 3 ∈ Odd )) | |
13 | oveq2 7143 | . . . . . 6 ⊢ (𝑞 = 3 → (3 + 𝑞) = (3 + 3)) | |
14 | 13 | eqeq2d 2809 | . . . . 5 ⊢ (𝑞 = 3 → (6 = (3 + 𝑞) ↔ 6 = (3 + 3))) |
15 | 11, 12, 14 | 3anbi123d 1433 | . . . 4 ⊢ (𝑞 = 3 → ((3 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (3 + 𝑞)) ↔ (3 ∈ Odd ∧ 3 ∈ Odd ∧ 6 = (3 + 3)))) |
16 | 10, 15 | rspc2ev 3583 | . . 3 ⊢ ((3 ∈ ℙ ∧ 3 ∈ ℙ ∧ (3 ∈ Odd ∧ 3 ∈ Odd ∧ 6 = (3 + 3))) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (𝑝 + 𝑞))) |
17 | 2, 2, 5, 16 | mp3an 1458 | . 2 ⊢ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (𝑝 + 𝑞)) |
18 | isgbe 44269 | . 2 ⊢ (6 ∈ GoldbachEven ↔ (6 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 6 = (𝑝 + 𝑞)))) | |
19 | 1, 17, 18 | mpbir2an 710 | 1 ⊢ 6 ∈ GoldbachEven |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 (class class class)co 7135 + caddc 10529 3c3 11681 6c6 11684 ℙcprime 16005 Even ceven 44142 Odd codd 44143 GoldbachEven cgbe 44263 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-prm 16006 df-even 44144 df-odd 44145 df-gbe 44266 |
This theorem is referenced by: nnsum3primesle9 44312 |
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